Nontrivial Solutions for a System of Second-Order Discrete Boundary Value Problems

Nonlinear discrete problems appear in many mathematical models, such as computer science, mechanical engineering, control systems, economics, and fluid mechanics (see [1–4]). Owing to the wide applications, in recent years, there are a large number of researchers paying special attention in this direction (we refer to some results [5–15] and the references therein). For example, in [5], the authors used the Guo–Krasnosel’skii fixed point theorem to study the existence of positive solutions for the following second-order discrete boundary value problem:


Introduction
Nonlinear discrete problems appear in many mathematical models, such as computer science, mechanical engineering, control systems, economics, and fluid mechanics (see [1][2][3][4]). Owing to the wide applications, in recent years, there are a large number of researchers paying special attention in this direction (we refer to some results [5][6][7][8][9][10][11][12][13][14][15] and the references therein). For example, in [5], the authors used the Guo-Krasnosel'skii fixed point theorem to study the existence of positive solutions for the following second-order discrete boundary value problem: and the following discrete second-order system: Δ 2 y i− 1 + g x i , y j � 0, i ∈ [1, n], where n is a positive integer, [1, n] � 1, 2, . . . , n { }, Δ is the forward difference operator, i.e., In [6], the authors used the monotone iterative technique to investigate the existence and uniqueness of positive solutions for the following discrete p-Laplacian fractional boundary value problem: where ] ∈ (0, 1) is a real number, Δ ] ]− 1 is a discrete fractional operator, and ϕ p (s) � |s| p− 2 s is the p-Laplacian with s ∈ R, p > 1.
In [12], the authors used the fixed point index to study the positive solutions for the following system of first-order discrete fractional boundary value problems: By discrete Jensen's inequality, the authors adopted some appropriate nonnegative concave and convex functions to characterize the coupling behavior of the nonlinearities f i (i � 1, 2).
Motivated by the aforementioned works, in this paper, by means of the topological degree theory, we study the existence of nontrivial solutions for the following system of second-order discrete boundary value problems: ) are continuous and satisfy the following conditions: where Now, we state our main result here. Theorem 1. Suppose that (H1)-(H4) hold. en, (6) has at least one nontrivial solution.

Preliminaries
Let E be the Banach space of real valued functions defined on the discrete interval T 2 with the norm ‖u‖ � max k∈T 2 |u(k)|, where T 2 : � 0, 1, 2, . . . , T + 1 { }. Define the following sets: and B r � x ∈ E: ‖x‖ < r { } for r > 0. en, P, P 0 are cones on E, and B r is an open ball in E.
Lemma 1 (see [11,15]). Let h(k) ∈ C(T 1 ). en, the discrete boundary value problem has a solution with the form where Furthermore, G(k, l) has the following properties (see [13,15]): By Lemma 1, system (6) is equivalent to en, we can define operators T, S: E ⟶ E by and operator A: Note that T, S, A are completely continuous operators (see [11]), and (u, v) solves (6) if and only if (u, v) is a fixed point of the operator A.
is is a direct result by Lemma 1 (ii), so we omit the proof.

Lemma 4 (see [25, eorem A.3.3]). Let Ω be a bounded open set in a Banach space E and T: Ω ⟶ E be a continuous compact operator. If there exists
then the topological degree deg(I − T, Ω, 0) � 0.

Lemma 5 (see [25, Lemma 2.5.1]). Let Ω be a bounded open set in a Banach space E with 0 ∈ Ω and T: Ω ⟶ E be a continuous compact operator. If
then the topological degree deg(I − T, Ω, 0) � 1.

Main Results
In order to obtain the Proof of eorem 1, we first provide a lemma.
So, from Lemma 3 and Remark 1, we have Note that u, v ∈ zB R , and using (24), R > N 1 , and R > N 2 , we have It is noted that ‖u‖ � ‖v‖ � R, u + u + v ∈ P 0 , and v + u + v ∈ P 0 . erefore, we get Using R > N 3 , we have and R > N 4 implies that